Is it possible that the model is overfitting when the training and validation accuracy increase?
I'll try to answer on more general questions

Is it ok that model performs better on validation, then on train?

It's certainly fine if you use techniques like dropout or data augmentation and the difference is not that big. Because in case of dropout for train you use part of the network, and for validation the whole.

I'm suspicious my model is too good. What could I do?

That a good point, because in my experience too good results sometime are reasoned by the flow in the training. The most common is data leakage, that's mean you give the model a way to fool around in unexpected and unwanted way.
Let me give an example. Lets imagine you try to detect malicious traffic and you have a data about sessions and you split records on train/val. Let's imagine one of the features is IP address and you could have multiple sessions from same IP. But if same IP would be in both train and val for the crooked users, then model could just remember the 'bad' IP and thus getting high score on both train and val.
So, to avoid it you need to think about things like that. If records could be group in some too meaningful way, make sure you put the whole group either in train or val. Another thing, that's it's usually good thing to split it by time as well, i.e. use for train records for 2018 and 2019 and for val for 2020. Thus you not only avoid data leakage, but you make sure your model is robust for future predictions.
More general, try to understand, why you getting such good result with such simple model. Try to squeeze the data even smaller, drop some features to understand, when it would go done.
Finally I would recommend the post by Andrej Karpathy, when he gave lots of really good advices on practical side of training NN. http://karpathy.github.io/2019/04/25/recipe/